^{1, 2}

^{2}

^{1}

^{2}

The traveling wave solutions and multiwave solutions to (3 + 1)-dimensional Jimbo-Miwa equation are investigated in this paper. As a result, besides the
exact bounded solitary wave solutions, we obtain the existence of two families
of bounded periodic traveling wave solutions and their implicit formulas by
analysis of phase portrait of the corresponding traveling wave system. We
derive the exact 2-wave solutions and two families of arbitrary finite

Various nonlinear partial differential equations (NLPDEs) have been proposed to model different kinds of phenomena in natural and applied sciences such as fluid dynamics, plasma physics, solid-state physics, optical fibers, acoustics, mechanics, biology, and mathematical finance. Obviously, it is of significant importance to study the solutions of such NLPDEs from both theoretical and practical points of view. However, the solution spaces of nonlinear equations are infinite-dimensional and contain diverse solution structures, so it is usually a difficult job to determine the solutions to nonlinear NLPDEs.

A great idea to generate exact solutions of NLPDEs is to reduce the NLPDEs into some algebraic equations by assuming the solutions to have some special forms or satisfy some solvable simpler equations. This can be seen in, for example, the exp-function method [

Recently, the planar dynamical system theorem has been employed to study the traveling wave solutions of NLPDEs [

The

The

In this paper, we firstly study the bounded traveling wave solutions of the Jimbo-Miwa equation (

To investigate the traveling wave solutions to the

First, we rewrite (

However, it is not easy to know the properties and the shapes of (

When

By further simplification, (

Thus, we have the following lemma.

The general second-order ODE (

According to the analysis and results in Section

Clearly,

However, we may not get bounded solutions from the family of periodic solutions of (

By the theory of planar dynamical system, when

The

The

where

The

In this section we study the

Let wave variables

Let

It follows from Theorem

Recall that the goal of this paper is to investigate the multiwave solutions to the

Let

To get other

For the case

That is to say, we get the traveling wave solutions to the

By further computation, (

To obtain the 2-wave solutions to the

From (

Let

To get the

Following the idea in [

For the case when

For any arbitrary positive integer

For the case when

Consequently, besides the family of

For any arbitrary positive integer

The dynamical system theory was employed to study the traveling wave solutions of the

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Nature Science Foundation of China (no. 11101371). This paper is motivated by nice discussion with Professor Wen-Xiu Ma during the authors’ visit at University of South Florida in March 2013. The authors would like to express their sincere appreciations to Professor Ma and the anonymous reviewers.